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5.4 Applying principle 7 to determine parallelograms Name the parallelograms in each part of Fig. 5-11.
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CHAPTER 5 Parallelograms, Trapezoids, Medians, and Midpoints
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Fig. 5-11
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(a) ABCD, AECF; (b) ABFD, BCDE; (c) ABDC, CDFE, ABFE.
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5.5 Applying principles 9, 10, and 11 State why ABCD is a parallelogram in each part of Fig. 5-12.
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Fig. 5-12
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(a) Since supplements of congruent angles are congruent, opposite angles of ABCD are congruent. Thus by Principle 10, ABCD is a parallelogram. (b) Since perpendiculars to the same line are parallel, AB i CD. Hence by Principle 9, ABCD is a parallelogram. (c) By the addition axiom, DG > GB. Hence by Principle 11, ABCD is a parallelogram.
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Proving a parallelogram problem
Given: ~ABCD E is midpoint of BC. G is midpoint of AD. EF ' BD, GH ' BD To Prove: EF > GH Plan: Prove nBFE > nGHD
PROOF:
Statements 1. E is midpoint of BC. G is midpoint of AD. 2. BE 1BC, GC 1AD 2 2 3. ABCD is a ~. 4. BC > AD 5. BE > GD 6. EF ' BD, GH ' BD 7. /1 > /2 8. BC i AD 9. /3 > /4 10. nBFE > nGHD 11. EF > GH 1. Given 2. 3. 4. 5. 6. 7. 8. 9. 10. 11.
Reasons
A midpoint cuts a segment in half. Given Opposite sides of a ~ are congruent. Halves of equals are equal. Given Perpendiculars form rt. . Rt. are >. Opposite sides of a ~ are i. Alternate interior of i lines are >. SAA s Corresponding parts of congruent ^ are >.
CHAPTER 5 Parallelograms, Trapezoids, Medians, and Midpoints
5.3 Special Parallelograms: Rectangle, Rhombus, and Square
5.3A Definitions and Relationships among the Special Parallelograms
Rectangles, rhombuses, and squares belong to the set of parallelograms. Each of these may be defined as a parallelogram, as follows: 1. A rectangle is an equiangular parallelogram. 2. A rhombus is an equilateral parallelogram. 3. A square is an equilateral and equiangular parallelogram. Thus, a square is both a rectangle and a rhombus. The relations among the special parallelograms can be pictured by using a circle to represent each set. Note the following in Fig. 5-13:
Fig. 5-13
1. Since every rectangle and every rhombus must be a parallelogram, the circle for the set of rectangles and the circle for the set of rhombuses must be inside the circle for the set of parallelograms. 2. Since every square is both a rectangle and a rhombus, the overlapping shaded section must represent the set of squares.
5.3B Principles Involving Properties of the Special Parallelograms
PRINCIPLE 1: PRINCIPLE 2: PRINCIPLE 3:
A rectangle, rhombus, or square has all the properties of a parallelogram. Each angle of a rectangle is a right angle. The diagonals of a rectangle are congruent.
Thus in rectangle ABCD in Fig. 5-14, AC > BD.
PRINCIPLE 4:
All sides of a rhombus are congruent.
Fig. 5-14
Fig. 5-15
CHAPTER 5 Parallelograms, Trapezoids, Medians, and Midpoints
PRINCIPLE 5:
The diagonals of a rhombus are perpendicular bisectors of each other.
Thus in rhombus ABCD in Fig. 5-15, AC and BD are bisectors of each other.
PRINCIPLE 6:
The diagonals of a rhombus bisect the vertex angles.
Thus in rhombus ABCD, AC bisects /A and /C.
PRINCIPLE 7:
The diagonals of a rhombus form four congruent triangles.
Thus in rhombus ABCD, nI > nII > nIII > nIV.
PRINCIPLE 8:
A square has all the properties of both the rhombus and the rectangle.
By definition, a square is both a rectangle and a rhombus.
5.3C Diagonal Properties of Parallelograms, Rectangles, Rhombuses, and Squares
Each check in the following table indicates a diagonal property of the figure. Diagonal Properties Diagonals bisect each other. Diagonals are congruent. Diagonals are perpendicular. Diagonals bisect vertex angles. Diagonals form 2 pairs of congruent triangles. Diagonals form 4 congruent triangles. Parallelogram Rectangle Rhombus Square
5.3D Proving that a Parallelogram is a Rectangle, Rhombus, or Square
Proving that a Parallelogram is a Rectangle
The basic or minimum definition of a rectangle is this: A rectangle is a parallelogram having one right angle. Since the consecutive angles of a parallelogram are supplementary, if one angle is a right angle, the remaining angles must be right angles. The converse of this basic definition provides a useful method of proving that a parallelogram is a rectangle, as follows:
PRINCIPLE 9:
If a parallelogram has one right angle, then it is a rectangle.
90 , then ABCD is a rectangle.
Thus if ABCD in Fig. 5-16 is a ~ and m/A
Fig. 5-16
PRINCIPLE 10:
If a parallelogram has congruent diagonals, then it is a rectangle.
Thus if ABCD is a ~ and AC > BD, then ABCD is a rectangle.
Proving that a Parallelogram is a Rhombus
The basic or minimum definition of a rhombus is this: A rhombus is a parallelogram having two congruent adjacent sides. The converse of this basic definition provides a useful method of proving that a parallelogram is a rhombus, as follows:
PRINCIPLE 11:
If a parallelogram has congruent adjacent sides, then it is a rhombus.
Thus if ABCD in Fig. 5-17 is a ~ and AB > BC, then ABCD is a rhombus.
CHAPTER 5 Parallelograms, Trapezoids, Medians, and Midpoints
Fig. 5-17
Proving that a Parallelogram is a Square
PRINCIPLE 12:
If a parallelogram has a right angle and two congruent adjacent sides, then it is a square.
This follows from the fact that a square is both a rectangle and a rhombus.
SOLVED PROBLEMS
5.7 Applying algebra to the rhombus Assuming ABCD is a rhombus, find x and y in each part of Fig. 5-18.
Fig. 5-18
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